Uncertainty Quantification in Linear Interpolation for Isosurface - - PowerPoint PPT Presentation
Uncertainty Quantification in Linear Interpolation for Isosurface Extraction Tushar Athawale and Alireza Entezari Department of Computer & Information Science & Engineering University of Florida Isosurface Visualization Figure:
Tushar Athawale and Alireza Entezari Department of Computer & Information Science & Engineering University of Florida
Figure: Isosurface representing the isovalue of 0 ◦C in a temperature
is courtesy of DEMETER project [Palmer et al., 2004]
(a) Isosurface Visualization (b) Positional Uncertainties
◮ Uncertainties or errors are introduced in various phases of the visualization pipeline
(from data acquisition until the final visualization), e.g., measurement errors.
◮ Uncertainties or errors are introduced in various phases of the visualization pipeline
(from data acquisition until the final visualization), e.g., measurement errors.
◮ Quantification and visualization of the uncertainties has become an important
reasearch direction.
◮ Uncertainties or errors are introduced in various phases of the visualization pipeline
(from data acquisition until the final visualization), e.g., measurement errors.
◮ Quantification and visualization of the uncertainties - important reasearch
direction.
◮ We study the effect of uncertain data on the marching cubes algorithm (MCA)
used for isosurface visualization [Lorensen and Cline, 1987].
◮ Cell Configuration Uncertainties. ◮ Geometric Uncertainties.
isovalue = 30
isovalue = 30
◮ Glyphs for flow field uncertainty visualization [Wittenbrink et al., 1996]. ◮ Color and opacity mapping [Rhodes et al., 2003]. ◮ Primitive displacement in the surface normal direction proportional to the
uncertainty [Grigoryan and Rheingans, 2004].
◮ Animation techniques, e.g., animated visual vibrations [Brown, 2004], probabilistic
animation [Lundstrom et al., 2007].
◮ Isosurface condition analysis to visualize the regions of isosurface, which are
sensitive to small data changes [P¨
◮ Visualization of anisotropic correlation structures to study structural variability in
level sets [Pfaffelmoser and Westermann, 2012].
◮ Choice of Gaussian process regression over trilinear interpolation when data
uncertainty is modeled using additive Gaussian noise [Schlegel et al., 2012].
Direct volume rendering of the probabilities of the level set crossing the cells, aka, level-crossing probabilities (LCP).
◮ Monte-Carlo sampling from Gaussian, non-parametric distributions [P¨
Hege, 2013].
◮
Approximate level-crossing probability (LCP) =
# samples that cross the isosurface #samples
. derived using Monte-Carlo approach
◮ Motivated by the work of P¨
probability density function.
◮ Motivated by the work of P¨
probability density function.
◮ Motivated by the work of P¨
probability density function.
◮ Motivated by the work of P¨
probability density function.
◮ We obtain analytic density function when data uncertainty is modeled using
uniform or kernel-based non-parametric distributions.
◮ Motivated by the work of P¨
probability density function.
◮ We obtain analytic density function when data uncertainty is modeled using
uniform or kernel-based non-parametric distributions.
◮ Closed-form characterization is efficient.
The level-crossing location for isovalue c, vc, is approximated using the inverse linear interpolation formula, vc = (1 − z)v1 + zv2, where z = c − x1 x2 − x1 .
Aim : Closed-form characterization of the ratio random variable, Z =
c−X1 X2−X1 , assuming X1
and X2 have uniform distributions.
µi and δi represent mean and width, respectively, of a random variable Xi. c is the
Find the joint distribution of the dependent random variables Z1 = c − X1 and Z2 = X2 − X1, where
Z2 = c−X1 X2−X1.
◮ Determine the range of c − X1. ◮ X1 assumes values in the range
[µ1 − δ1, µ1 + δ1].
◮ Random variables Z1 and Z2 are
dependent. µi and δi represent mean and width, respectively, of a random variable Xi.
◮ Determine the range of X2 − X1. ◮ X2 assumes values in the range
[µ2 − δ2, µ2 + δ2].
◮ Random variables Z1 and Z2 are
dependent. µi and δi represent mean and width, respectively, of a random variable Xi.
◮ Determine the range of X2 − X1. ◮ X2 assumes values in the range
[µ2 − δ2, µ2 + δ2].
◮ Random variables Z1 and Z2 are
dependent. µi and δi represent mean and width, respectively, of a random variable Xi.
◮ Parallelogram represents the
joint distribution of the dependent random variables Z1 = c − X1 and Z2 = X2 − X1.
◮ Parallelogram represents the
joint distribution of the dependent random variables Z1 = c − X1 and Z2 = X2 − X1.
◮ Uniform kernel: parallelogram
with constant height.
◮ Parallelogram represents the
joint distribution of the dependent random variables Z1 = c − X1 and Z2 = X2 − X1.
◮ Uniform kernel: parallelogram
with constant height.
◮ Parzen window, triangular
kernels: parallelogram with height described by a polynomial function.
Shape and position of the joint distribution is impacted by the relative configurations for X1 and X2 and the isovalue c.
(a) Non-overlapping (b) Overlapping (c) Contained
What is Pr( Z1
Z2 ≤ m)?
◮ What is Pr( Z1 Z2 ≤ m)? ◮ cdfZ(m) = Pr(−∞ ≤ Z1 Z2 ≤ m)
(orange region). cdfZ(m) represents cumulative density function of a random variable Z.
◮ What is Pr( Z1 Z2 ≤ m)? ◮ cdfZ(m) = Pr(−∞ ≤ Z1 Z2 ≤ m)
(orange region).
◮ Obtain pdfZ(m) by differentiating
cdfZ(m) with respect to m. pdfZ(m) represents probability density function of a random variable Z.
◮ What is Pr( Z1 Z2 ≤ m)? ◮ cdfZ(m) = Pr(−∞ ≤ Z1 Z2 ≤ m)
(orange region).
◮ Obtain pdfZ(m) by differentiating
cdfZ(m) with respect to m.
◮ A piecewise function.
◮ What is Pr( Z1 Z2 ≤ m)? ◮ cdfZ(m) = Pr(−∞ ≤ Z1 Z2 ≤ m)
(orange region).
◮ Obtain pdfZ(m) by differentiating
cdfZ(m) with respect to m.
◮ A piecewise function. ◮ Each piece is an inverse polynomial.
2
4δ1δ2(1−m)2
2
4δ1δ2(1−m)2
8δ1δ2m2(1−m)2
8δ1δ2m2(1−m)2
1
4δ1δ2m2
1
4δ1δ2m2
8δ1δ2m2(1−m)2
8δ1δ2m2(1−m)2
2
4δ1δ2(1−m)2
We get a piecewise density function as follows, where each piece is an inverse polynomial: pdfZ(m) =
(c−µ2)2+δ2
2
4δ1δ2(1−m)2 ,
−∞ < m ≤ slope S.
(µ2+δ2−c)2m2+(µ1+δ1−c)2(1−m)2 8δ1δ2m2(1−m)2
, slope S < m ≤ slope Q.
(c−µ1)2+δ2
1
4δ1δ2m2
, slope Q < m ≤ slope P.
(µ2+δ2−c)2m2+(µ1−δ1−c)2(1−m)2 8δ1δ2m2(1−m)2
, slope P < m ≤ slope R.
(c−µ2)2+δ2
2
4δ1δ2(1−m)2 ,
slope R < m < ∞.
◮ Determine cell edges that are getting crossed by the isosurface using MCA
[Lorensen and Cline, 1987].
◮ Determine cell edges that are getting crossed by the isosurface using MCA
[Lorensen and Cline, 1987].
◮ Find the edge-crossing density function for the edges that are crossed by the
isosurface (conditional probability).
(d) Non-overlapping (e) Overlapping (f) Contained Figure: Exemplar probability distribution functions, limited to the domain [0,1], for various interval cases. From left to right: non-overlapping (µ1 = 3, δ1 = 3, µ2 = 12, δ2 = 4, and c = 7.8), overlapping (µ1 = 5, δ1 = 4, µ2 = 12, δ2 = 6, and c = 8.8) and contained intervals (µ1 = 7, δ1 = 2, µ2 = 8, δ2 = 6, and c = 4.9).
◮ Determine cell edges that are getting crossed by the isosurface using MCA
[Lorensen and Cline, 1987].
◮ Find the edge-crossing density function for the edges that are crossed by the
isosurface (conditional probability).
◮ Find expected crossing location and quantify its uncertainty using the variance.
◮ Determine cell edges that are getting crossed by the isosurface using MCA
[Lorensen and Cline, 1987].
◮ Find the edge-crossing density function for the edges that are crossed by the
isosurface (conditional probability).
◮ Find expected crossing location and quantify its uncertainty using variance. ◮ Analytic edge-crossing density function allows closed-form computations of the
expected value and variance.
(a) Ground Truth (b) Linear Interpolation Isosurface (c) Expected Isosurface (d) Positional Uncertainties
Ground Truth Linear Interpolation Isosurface Expected Isosurface Positional Uncertainties
◮ Level-crossing probabilities (LCP) for each cell can be computed in closed form. ◮ Volume rendering of the LCP.
◮ Analytic formulation of the edge-crossing probability density function for the
kernels with compact support.
◮ Efficient stable isosurface reconstruction (with closed-form expected value
computation) from uncertain data.
◮ Visualization of the positional uncertainties in the expected isosurface (with
closed-form variance computation).
This research is supported by the AFOSR grant FA9550-12-1-0304, NSF grant CCF-1018149, and ONR grant N000141210862.